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Convex Cocompact Actions in Real Projective Geometry

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Convex Cocompact Actions in Real Projective Geometry CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL Abstract. We study a notion of convex cocompactness for (not nec- essarily irreducible) discrete subgroups of the projective general linear group acting on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good properties of classical con- vex cocompact subgroups in rank-one Lie groups. Extending our earlier work [DGK3] from the context of projective orthogonal groups, we show that for word hyperbolic groups preserving a properly convex open set in projective space, the above general notion of convex cocompactness is equivalent to a stronger convex cocompactness condition studied by Crampon–Marquis, and also to the condition that the natural inclusion be a projective Anosov representation. We investigate examples. Contents 1. Introduction 2 2. Reminders 16 3. Basic facts: actions on convex subsets of P(V ) 19 4. Convex sets with bisaturated boundary 23 5. Duality for convex cocompact actions 30 6. Segments in the full orbital limit set 33 7. Convex cocompactness and no segment implies P1-Anosov 38 8. P1-Anosov implies convex cocompactness 40 9. Smoothing out the nonideal boundary 44 10. Properties of convex cocompact groups 47 p;q−1 11. Convex cocompactness in H 54 J.D. was partially supported by an Alfred P. Sloan Foundation fellowship, and by the National Science Foundation under grant DMS 1510254. F.G. and F.K. were partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11- LABX-0007-01) and the grant DynGeo (ANR-16-CE40-0025-01). F.K. was partially sup- ported by the European Research Council under ERC Starting Grant 715982 (DiGGeS). The authors also acknowledge support from the GEAR Network, funded by the National Science Foundation under grants DMS 1107452, 1107263, and 1107367 (“RNMS: GEomet- ric structures And Representation varieties”). Part of this work was completed while F.K. was in residence at the MSRI in Berkeley, California, for the program Geometric Group Theory (Fall 2016) supported by NSF grant DMS 1440140, and at the INI in Cambridge, UK, for the program Nonpositive curvature group actions and cohomology (Spring 2017) supported by EPSRC grant EP/K032208/1. 1 2 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL 12. Examples of groups which are convex cocompact in P(V ) 67 Appendix A. Some open questions 73 References 74 1. Introduction In the classical setting of semisimple Lie groups G of real rank one, a discrete subgroup of G is said to be convex cocompact if it acts cocompactly on some nonempty closed convex subset of the Riemannian symmetric space G=K of G. Such subgroups have been abundantly studied, in particular in the context of Kleinian groups and real hyperbolic geometry, where there is a rich world of examples. They are known to display good geometric and dynamical behavior. On the other hand, in higher-rank semisimple Lie groups G, the condition that a discrete subgroup Γ act cocompactly on some nonempty convex subset of the Riemannian symmetric space G=K turns out to be quite restrictive: Kleiner–Leeb [KL] and Quint [Q] proved, for example, that if G is simple and such a subgroup Γ is Zariski-dense in G, then it is in fact a uniform lattice of G. The notion of an Anosov representation of a word hyperbolic group in a higher-rank semisimple Lie group G, introduced by Labourie [L] and gener- alized by Guichard–Wienhard [GW3], is a much more flexible notion which has earned a central role in higher Teichmüller–Thurston theory, see e.g. [BIW2, BCLS, KLPc, GGKW, BPS]. Anosov representations are defined, not in terms of convex subsets of the Riemannian symmetric space G=K, but instead in terms of a dynamical condition for the action on a certain flag variety, i.e. on a compact homogeneous space G=P . This dynamical condi- tion guarantees many desirable analogies with convex cocompact subgroups in rank one: see e.g. [L, GW3, KLPa, KLPb, KLPc]. It also allows for the definition of certain interesting geometric structures associated to Anosov representations: see e.g. [GW1, GW3, KLPb, GGKW, CTT]. However, nat- ural convex geometric structures associated to Anosov representations have been lacking in general. Such structures could allow geometric intuition to bear more fully on Anosov representations, making them more accessible through familiar geometric constructions such as convex fundamental do- mains, and potentially unlocking new sources of examples. While there is a rich supply of examples of Anosov representations into higher-rank Lie groups in the case of surface groups or free groups, it has proven difficult to construct examples for more complicated word hyperbolic groups. n One of the goals of this paper is to show that, when G = PGL(R ) is a projective linear group, there are in many cases natural convex cocompact n geometric structures modeled on P(R ) associated to Anosov representations n into G. In particular, we prove that any Anosov representation into PGL(R ) which preserves a nonempty properly convex open subset of the projective CONVEX COCOMPACT ACTIONS IN REAL PROJECTIVE GEOMETRY 3 n space P(R ) satisfies a strong notion of convex cocompactness introduced by Crampon–Marquis [CM]. Conversely, we show that convex cocompact sub- n groups of P(R ) in the sense of [CM] always give rise to Anosov representa- tions, which enables us to give new examples of Anosov representations and study their deformation spaces by constructing these geometric structures directly. In [DGK3] we had previously established this close connection be- tween convex cocompactness in projective space and Anosov representations in the case of irreducible representations valued in a projective orthogonal group PO(p; q). One context where a connection between Anosov representations and con- vex projective structures has been known for some time is the deformation 3 theory of real projective surfaces, for G = PGL(R ) [Go, CGo]. More gen- erally, it follows from work of Benoist [B3] that if a discrete subgroup Γ of n G = PGL(R ) divides (i.e. acts cocompactly on) a strictly convex open sub- n set of P(R ), then Γ is word hyperbolic and the natural inclusion Γ ,! G is Anosov. n Benoist [B6] also found examples of discrete subgroups of PGL(R ) which divide properly convex open sets that are not strictly convex; these sub- groups are not word hyperbolic. In this paper we study a notion of convex n n cocompactness for discrete subgroups of PGL(R ) acting on P(R ) which simultaneously generalizes Crampon–Marquis’s notion and Benoist’s convex divisible sets [B3, B4, B5, B6]. In particular, we show that this notion is stable under deformation into larger projective general linear groups, after the model of quasi-Fuchsian deformations of Fuchsian groups. This yields n examples of nonhyperbolic irreducible discrete subgroups of PGL(R ) which n are convex cocompact in P(R ) but do not divide a properly convex open n subset of P(R ). We now describe our results in more detail. 1.1. Strong convex cocompactness in P(V ) and Anosov representa- n tions. In the whole paper, we fix an integer n ≥ 2 and set V := R . Recall that an open domain Ω in the projective space P(V ) is said to be properly convex if it is convex and bounded in some affine chart, and strictly convex if in addition its boundary does not contain any nontrivial projective line segment. It is said to have C1 boundary if every point of the boundary of Ω has a unique supporting hyperplane. In [CM], Crampon–Marquis introduced a notion of geometrically finite action of a discrete subgroup Γ of PGL(V ) on a strictly convex open domain 1 of P(V ) with C boundary. If cusps are not allowed, this notion reduces to a natural notion of convex cocompact action on such domains. We will call discrete groups Γ with such actions strongly convex cocompact. Definition 1.1. Let Γ ⊂ PGL(V ) be an infinite discrete subgroup. • Let Ω be a Γ-invariant properly convex open subset of P(V ) whose boundary is strictly convex and C1. The action of Γ on Ω is convex orb cocompact if the convex hull in Ω of the orbital limit set ΛΩ (Γ) of 4 JEFFREY DANCIGER, FRANÇOIS GUÉRITAUD, AND FANNY KASSEL Γ in Ω is nonempty and has compact quotient by Γ. (For convex cocompact actions on non-strictly convex Ω, see Section 1.4.) • The group Γ is strongly convex cocompact in P(V ) if it admits a convex cocompact action on some nonempty properly convex open 1 subset Ω of P(V ) whose boundary is strictly convex and C . orb Here the orbital limit set ΛΩ (Γ) of Γ in Ω is defined as the set of accu- mulation points in @Ω of the Γ-orbit of some point z 2 Ω, and the convex orb orb hull of ΛΩ (Γ) in Ω as the intersection of Ω with the convex hull of ΛΩ (Γ) orb in the convex set Ω. It is easy to see that ΛΩ (Γ) does not depend on the choice of z since Ω is strictly convex. We deduce the following remark. Remark 1.2. The action of Γ on the strictly convex open set Ω is convex cocompact in the sense of Definition 1.1 if and only if Γ acts cocompactly on some nonempty closed convex subset of Ω. p;1 p Example 1.3. For V = R with p ≥ 2, any discrete subgroup of Isom(H ) = PO(p; 1) ⊂ PGL(V ) which is convex cocompact in the usual sense is strongly p convex cocompact in P(V ), taking Ω to be the projective model of H .
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